Final Project
Justin Wininger, Otis Murray, Henry Katz
2023-11-29
Introduction
This project explores the relationships between Total Yards Per Game (YPG) and the Points Per Game (PPG) that a team scores in both the National Football League (NFL), as well as the Division 1 Football Bowl Subdivision (FBS) of College football. These results are then compared to answer the question of whether PPG and YPG have a differing relationship between the NFL and FBS. This project first examines the individual relationships between the NFL teams’ YPG and their PPG, as well as examining how passing yards per game (PYPG) and rushing yards per game (RYPG) affect those results. This process is then repeated for the FBS, and then the resulting data is used to conclude how similar these relationships are in the FBS versus the more competitive NFL. This project heavily relies on simple linear regression (SLR), multiple linear regression (MLR), and analysis of covariance (ANCOVA) in order to display the differing trends between PPG and the YPG.
NFl Data
First, we set off to explore the NFL data set that we scraped from pro-football-reference.com. This data set included a table with all 32 NFL teams, as well as statistics such as total yards, total passing yards, total rushing yards, etc. We then proceeded to tidy this table into one that only involved the Team, PPG, YPG, PYPG, and RYPG. Using this table we then created scatter plots for the three relationships: YPG v.s. PPG, PYPG v.s. PPG, and RYPG v.s. PPG. Using these scatter plots we will analyze using SLR, MLR, and finally ANCOVA.
NFL Visualizations
Below is a table that attempts to show a relationship between PPG and YPG in the NFL.
| Team | YPG Rank | PPG Rank | |
|---|---|---|---|
| 10 | Chiefs | 1 | 1 |
| 4 | Bills | 2 | 2 |
| 15 | Eagles | 3 | 3 |
| 13 | Cowboys | 11 | 4 |
| 20 | Lions | 4 | 5 |
| 1 | 49ers | 5 | 6 |
| 3 | Bengals | 8 | 7 |
| 32 | Vikings | 7 | 8 |
| 28 | Seahawks | 13 | 9 |
| 18 | Jaguars | 10 | 10 |
| 14 | Dolphins | 6 | 11 |
| 24 | Raiders | 12 | 12 |
| 9 | Chargers | 9 | 13 |
| 21 | Packers | 17 | 14 |
In this table, the top 14 NFL teams are sorted by their rank (out of 32) in PPG in ascending order. Their rank in YPG is also shown. On the far left is each team’s ranking alphabetically. This is done to establish a baseline. This table attempts to show that, in general, the teams ranked highly in PPG are also ranked highly in YPG. For example, the top three teams in PPG, the Kansas City Chiefs, Buffalo Bills, and Philadelphia Eagles respectively, are also the top three teams in YPG, in the same order. The fourth ranked team in PPG, the Dallas Cowboys is only 11th in YPG, so the pattern is not true for all 32 teams. However, of the top 10 teams in PPG, only two are outside the top 10 in YPG, the aforementioned Cowboys and the Seattle Seahawks. Those two teams are ranked 11th and 13th in YPG respectively. The fact that the top 10 teams in PPG are all top 13 in YPG is an excellent indicator that there may be a relationship between PPG and YPG. We will conduct further analysis to see whether there really is a relationship between PPG and YPG or not.
Going further into that exploration, below are three scatter plots. The first shows the relationship between PPG and YPG in the NFL. The second shows the relationship between PPG and PYPG in the NFL. The third shows the relationship between PPG and RYPG in the NFL.
## Warning: Using the `size` aesthetic with geom_line was deprecated in ggplot2 3.4.0.
## ℹ Please use the `linewidth` aesthetic instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
All three scatter plots show a positive, linear relationship. The first two, PPG vs YPG and PPG vs PYPG respectively, show a very clear and obvious positive linear relationship. The third plot, PPG vs RYPG, shows a less clear and obvious positive linear relationship. There appears to be some correlation, but not a lot. Regardless, the most important thing is that y increases as x increases. This is true for all three plots, whether x is YPG, PYPG, or RYPG.
NFL Analysis
Simple Linear Regression
We decided to perform simple linear regression to see if PPG and YPG are related. In order to infer, some conditions must be met. The data must be linear, have equal variance, and must be normal. Below are two plots that asses whether or not the conditions are met.
## `geom_smooth()` using formula = 'y ~ x'
The first condition that must be met is linearity. The explanatory and response variables must have a linear relationship. This was proven by the first scatter plot of PPG and YPG. The linearity condition, therefore, is met. The second condition that must be met is equal variance. The first plot above is a scatter plot of residuals versus fitted values. The points in this plot must show an equal spread across the width. If they do, then the data have equal variance. Our residual vs fitted plot is fairly good. It is certainly not perfect, but nothing ever is in the real world. The plot shows enough spread and no obvious patterns, so the equal variance condition, therefore, is met. The third and final condition that must be met is normality. The data must be normally distributed. A Q-Q plot assesses this condition. If the Q-Q plot shows a strong linear relationship then the data are normal. Our QQ plot shows a relatively strong linear relationship. Like before, it is not perfect, but it is good enough. The normality condition, therefore, is met.
Because all the conditions for inference are met, we can now infer from the data. The PPG vs YPG model has a test statistic of \(9.842055\) which corresponds to a p-value of \(6.620226 × 10^{-11}\).
Multiple Linear Regression
We also decided to perform multiple linear regression to see how if PPG is related to PYPG and RYPG. The conditions for inference for multiple linear regression are the same as simple linear regression, with one additional condition. There must not be multicollinearity. Below are plots that asses the conditions for multiple linear regression.
## `geom_smooth()` using formula = 'y ~ x'
In multiple linear regression, the linearity condition is met if all predictors have a linear relationship. This linear relationship is seen in the second and third scatter plots from the visualization section. Those scatter plots prove that all predictors have a linear relationship, even if some are not that good. The linearity condition, therefore, is met. Unlike linearity, equal variance and normality are the same as before. The residual versus fitted plot shows a decent spread across the width, so the equal variance condition, therefore, is met. The Q-Q plot shows a weak linear relationship, but there still is one. We cannot say for sure if normality is met, but we will still proceed with caution as if it is met. Multicollinearity occurs if two independent variables have a high correlation. The two independent variables are the two explanatory variables, PYPG and RYPG. The third plot, a scatter plot that shows the relationship between PYPG and RYPG, should be a good indicator of if multicollinearity is present or not. If this scatter plot showed a strong relationship, it would be evidence that multicollinearity is present. It does not, however. Additionally, variance inflation factor (VIF) is a direct measure of multicollinearity. Generally, a value of five or above means multicollinearity is present. The VIF value for both predictors is \(1.421673\), so the multicollinearity condition, therefore, is met.
All conditions for inference are met. The multiple linear regression model has two predictors, and will therefore have two test statistics and p-values. For PYPG, the test statistic is \(9.814515\) which corresponds to a p-value of \(1.007533 × 10^{-10}\). For RYPG, the test statistic is \(7.282251\) which corresponds to a p-value of \(5.086612 × 10^{-8}\).
Regression Conclusion
All p-values are essentially zero. For the simple linear regression, this means that we have proven that PPG and YPG have a linear relationship. As YPG increases, so does PPG. For the multiple linear regression,this means that we can conclude that both predictors are statistically significant. Both RPYG and PYPG, therefore, significantly affect PPG.
College Data
#First, we set off to explore the NFL data set that we scraped from pro-football-reference.com. This data set included a table with all 32 NFL teams, as well as statistics such as total yards, total passing yards, total rushing yards, etc. We then proceeded to tidy this table into one that only involved the Team, PPG, YPG, PYPG, and RYPG. Using this table we then created scatter plots for the three relationships: YPG v.s. PPG, PYPG v.s. PPG, and RYPG v.s. PPG. Using these scatter plots we will analyze using SLR, MLR, and finally ANCOVA.
Below we examine the College Football data set that we wrangled from sports-reference.com. Unlike the NFL data set, likely due to the large difference in number of teams, this data set had far less unnecessary statistics that had to be removed in order to create a tided table. The data was then examined for the same relationships as the NFL data set: YPG v.s. PPG, PYPG v.s. PPG, and RYPG v.s. PPG. Further, we then created three more scatter plots and repeated the process of analyzing them using SLR, MLR, and ANCOVA.
College Visualizations
Below is a table that attempts to show a relationship between PPG and YPG in College.
| Team | YPG Rank | PPG Rank | |
|---|---|---|---|
| 103 | Tennessee | 1 | 1 |
| 80 | Ohio State | 9 | 2 |
| 116 | USC | 3 | 3 |
| 3 | Alabama | 11 | 4 |
| 35 | Georgia | 5 | 5 |
| 62 | Michigan | 24 | 6 |
| 125 | Washington | 2 | 7 |
| 114 | UCLA | 4 | 8 |
| 85 | Oregon | 6 | 9 |
| 101 | TCU | 27 | 10 |
| 117 | Utah | 17 | 11 |
| 94 | SMU | 14 | 12 |
| 45 | James Madison | 29 | 13 |
| 120 | UTSA | 12 | 14 |
This table is the exact same as the NFL table in how it is formatted. It shows the top 14 teams’ ranks in PPG in descending order.
## `geom_smooth()` using formula = 'y ~ x'
##
## Call:
## lm(formula = `Points Per Game` ~ `Yards Per Game`, data = collegeOffense)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.3760 -1.8875 0.3087 1.7450 5.6176
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.256685 1.528327 -8.674 1.54e-14 ***
## `Yards Per Game` 0.105643 0.003854 27.410 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.672 on 129 degrees of freedom
## Multiple R-squared: 0.8535, Adjusted R-squared: 0.8523
## F-statistic: 751.3 on 1 and 129 DF, p-value: < 2.2e-16## `geom_smooth()` using formula = 'y ~ x'
##
## Call:
## lm(formula = `Points Per Game` ~ `Passing Yards Per Game` + `Rushing Yards Per Game`,
## data = collegeOffense)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.3263 -1.8484 0.1267 1.7477 5.6455
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.262958 1.531178 -8.662 1.72e-14 ***
## `Passing Yards Per Game` 0.103847 0.004591 22.621 < 2e-16 ***
## `Rushing Yards Per Game` 0.108315 0.005343 20.272 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.677 on 128 degrees of freedom
## Multiple R-squared: 0.8541, Adjusted R-squared: 0.8518
## F-statistic: 374.5 on 2 and 128 DF, p-value: < 2.2e-16
## Analysis of Variance Table
##
## Response: Points Per Game
## Df Sum Sq Mean Sq F value Pr(>F)
## `Yards Per Game` 1 6683.0 6683.0 1044.5739 <2e-16 ***
## Level 1 14.1 14.1 2.1975 0.1402
## Residuals 160 1023.7 6.4
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Analysis of Variance Table
##
## Response: Points Per Game
## Df Sum Sq Mean Sq F value Pr(>F)
## `Passing Yards Per Game` 1 2910.9 2910.87 112.197 < 2.2e-16 ***
## Level 1 658.8 658.78 25.392 1.252e-06 ***
## Residuals 160 4151.1 25.94
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Analysis of Variance Table
##
## Response: Points Per Game
## Df Sum Sq Mean Sq F value Pr(>F)
## `Rushing Yards Per Game` 1 2427.4 2427.42 77.1248 2.34e-15 ***
## Level 1 257.5 257.51 8.1819 0.004796 **
## Residuals 160 5035.8 31.47
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Conclusion
narrative text here
Code Appendix
#load necessary packages
knitr::opts_chunk$set(echo = FALSE, dpi=300)
library("groundhog")
pkgs <- c('Stat2Data', 'tidyverse', 'mosaic', 'ggformula', 'Lock5Data', 'tinytex', 'car', 'leaps', 'HH', 'rvest', 'dplyr', 'readr', 'ggplot2', 'tidyr', 'readxl', 'esquisse', 'kableExtra', 'ggformula')
groundhog.library(pkgs, '2023-11-29') #Using the date when project was started
#create NFL table
nflTable <- read_html(
x = "http://pfref.com/pi/share/d7riA"
) %>%
html_elements(css = "table") %>%
html_table()
nflOffense <- nflTable[[1]]
#delete unnecessary data
nflOffense <- subset(nflOffense, select=-c(1, 6:12, 14:18, 20:28))
nflOffense <- nflOffense[-c(1, 22:23, 36:38), ]
#rename columns
nflOffense <- nflOffense%>%rename(
`Team`=1, `Games`=2, `Total Points`=3, `Total Yards`=4, `Total Passing Yards`=5, `Total Rushing Yards`=6)
#squish team names
nflOffense$Team <- sub(".*\\s", "", nflOffense$Team)
#rearrange column order
#nflOffense <- nflOffense[,c("Team", "Games","Points","Yards")]
#make every cell a number versus a character
nflOffense$Games <- as.numeric(nflOffense$Games)
nflOffense$`Total Points` <- as.numeric(nflOffense$`Total Points`)
nflOffense$`Total Yards` <- as.numeric(nflOffense$`Total Yards`)
nflOffense$`Total Passing Yards` <- as.numeric(nflOffense$`Total Passing Yards`)
nflOffense$`Total Rushing Yards` <- as.numeric(nflOffense$`Total Rushing Yards`)
#make per game stats
nflOffense$`Points Per Game` <- nflOffense$`Total Points` / nflOffense$Games
nflOffense$`Yards Per Game` <- nflOffense$`Total Yards` / nflOffense$Games
nflOffense$`Passing Yards Per Game` <- nflOffense$`Total Passing Yards` / nflOffense$Games
nflOffense$`Rushing Yards Per Game` <- nflOffense$`Total Rushing Yards` / nflOffense$Games
#remove unnecessary data
nflOffense <- subset(nflOffense, select=-c(2:6))
#round down to tenths place
nflOffense$`Points Per Game` <- round(nflOffense$`Points Per Game`, digits = 1)
nflOffense$`Yards Per Game` <- round(nflOffense$`Yards Per Game`, digits = 1)
nflOffense$`Passing Yards Per Game` <- round(nflOffense$`Passing Yards Per Game`, digits = 1)
nflOffense$`Rushing Yards Per Game` <- round(nflOffense$`Rushing Yards Per Game`, digits = 1)
#rearrange columns
nflOffense <- nflOffense[, c(1, 2, 4, 5, 3)]
#create new dataframe with just points per game
nflOffensePPG <- subset(nflOffense, select=-c(3:5))
#order the rows by PPG
nflOffensePPG <- nflOffensePPG[order(nflOffensePPG$`Points Per Game`, decreasing = TRUE),]
#create new dataframe with just yards per game
nflOffenseYPG <- subset(nflOffense, select=-c(2:4))
#order the rows by YPG
nflOffenseYPG <- nflOffenseYPG[order(nflOffenseYPG$`Yards Per Game`, decreasing = TRUE),]
nflOffensePPG <- nflOffensePPG%>%
mutate(`PPG Rank`= c(1:32))
nflOffensePPG <- subset(nflOffensePPG, select=-c(2))
nflOffenseYPG <- nflOffenseYPG%>%
mutate(`YPG Rank`= c(1:32))
nflOffenseYPG <- subset(nflOffenseYPG, select=-c(2))
nflRanks <- merge(nflOffenseYPG, nflOffensePPG, by = "Team", all = TRUE)
nflRanks <- nflRanks[order(nflRanks$`PPG`, decreasing = FALSE),]
nflRanks14 <- head(nflRanks, n=14)
nflRanks14%>%
kable(
caption="Every NFL team's rank in PPG and YPG",
booktabs=TRUE,
align=c("l", rep("c",6))
)%>%
kableExtra::kable_styling(
bootstrap_options=c("striped", "condensed"),
font_size=16
)
gf_point(`Points Per Game`~`Yards Per Game`,
data=nflOffense,
linewidth=1,
title= 'Points Per Game vs Yards Per Game in the NFL')%>%
gf_lm()
gf_point(`Points Per Game`~`Passing Yards Per Game`,
data=nflOffense,
linewidth=1,
title= 'Points Per Game vs Passing Yards Per Game in the NFL')%>%
gf_lm()
gf_point(`Points Per Game`~`Rushing Yards Per Game`,
data=nflOffense,
linewidth=1,
title= 'Points Per Game vs Rushing Yards Per Game in the NFL')%>%
gf_lm()
nflTotalYardsSLRModel <- lm(`Points Per Game`~`Yards Per Game`, data=nflOffense)
mplot(nflTotalYardsSLRModel, which=1)
mplot(nflTotalYardsSLRModel, which=2)
nflTotalYardsSLRSummary <- summary(nflTotalYardsSLRModel)
nflTotalYardsSLROutput <- nflTotalYardsSLRSummary[["coefficients"]]
nflMLRModel <- lm(`Points Per Game`~`Passing Yards Per Game`+`Rushing Yards Per Game`, data=nflOffense)
mplot(nflMLRModel, which=1)
mplot(nflMLRModel, which=2)
gf_point(`Passing Yards Per Game`~`Rushing Yards Per Game`, data=nflOffense)
vif <- vif(nflMLRModel)
nflMLRSummary<- summary(nflMLRModel)
nflMLROutput <- nflMLRSummary[["coefficients"]]
collegeTable <- read_html(
x = "https://www.sports-reference.com/cfb/years/2022-team-offense.html"
) %>%
html_elements(css = "table") %>%
html_table()
collegeOffense <- collegeTable[[1]]
collegeOffense <- subset(collegeOffense, select=-c(1, 3, 5:7, 9:10, 12:14, 16:25))
collegeOffense <- collegeOffense%>%rename(
`Team`=1, `Points Per Game`=2, `Passing Yards Per Game`=3, `Rushing Yards Per Game`=4, `Yards Per Game`=5
)
collegeOffense <- collegeOffense[-c(1, 22:23, 44:45, 66:67, 88:89, 110:111, 132:133),]
collegeOffense$`Points Per Game` <- as.numeric(collegeOffense$`Points Per Game`)
collegeOffense$`Passing Yards Per Game` <- as.numeric(collegeOffense$`Passing Yards Per Game`)
collegeOffense$`Rushing Yards Per Game` <- as.numeric(collegeOffense$`Rushing Yards Per Game`)
collegeOffense$`Yards Per Game` <- as.numeric(collegeOffense$`Yards Per Game`)
#create new dataframe with just points per game
collegeOffensePPG <- subset(collegeOffense, select=-c(3:5))
#order the rows by PPG
collegeOffensePPG <- collegeOffensePPG[order(collegeOffensePPG$`Points Per Game`, decreasing = TRUE),]
#create new dataframe with just yards per game
collegeOffenseYPG <- subset(collegeOffense, select=-c(2:4))
#order the rows by YPG
collegeOffenseYPG <- collegeOffenseYPG[order(collegeOffenseYPG$`Yards Per Game`, decreasing = TRUE),]
collegeOffensePPG <- collegeOffensePPG%>%
mutate(`PPG Rank`= c(1:131))
collegeOffensePPG <- subset(collegeOffensePPG, select=-c(2))
collegeOffenseYPG <- collegeOffenseYPG%>%
mutate(`YPG Rank`= c(1:131))
collegeOffenseYPG <- subset(collegeOffenseYPG, select=-c(2))
collegeRanks <- merge(collegeOffenseYPG, collegeOffensePPG, by = "Team", all = TRUE)
collegeRanks <- collegeRanks[order(collegeRanks$`PPG`, decreasing = FALSE),]
collegeRanks14 <- head(collegeRanks, n=14)
collegeRanks14%>%
kable(
caption="Every college team's rank in PPG and YPG",
booktabs=TRUE,
align=c("l", rep("c",6))
)%>%
kableExtra::kable_styling(
bootstrap_options=c("striped", "condensed"),
font_size=16,
)
gf_point(`Points Per Game`~`Yards Per Game`,
data=collegeOffense,
linewidth=1,
title= 'Points Per Game vs Yards Per Game in College')%>%
gf_lm()
gf_point(`Points Per Game`~`Passing Yards Per Game`,
data=collegeOffense,
linewidth=1,
title= 'Points Per Game vs Passing Yards Per Game in College')%>%
gf_lm()
gf_point(`Points Per Game`~`Rushing Yards Per Game`,
data=collegeOffense,
linewidth=1,
title= 'Points Per Game vs Rushing Yards Per Game in College')%>%
gf_lm()
collegeTotalYardsSLRModel <- lm(`Points Per Game`~`Yards Per Game`, data=collegeOffense)
collegeMLRModel <- lm(`Points Per Game`~`Passing Yards Per Game`+`Rushing Yards Per Game`, data=collegeOffense)
mplot(collegeTotalYardsSLRModel, which=1)
mplot(collegeTotalYardsSLRModel, which=2)
summary(collegeTotalYardsSLRModel)
mplot(collegeMLRModel, which=1)
mplot(collegeMLRModel, which=2)
summary(collegeMLRModel)
collegeOffense <- collegeOffense%>%
mutate(`Level`= "College")
nflOffense <- nflOffense%>%
mutate(`Level`= "NFL")
offense <- rbind(nflOffense,collegeOffense)
offense$IndLevel <- as.numeric(offense$Level=="NFL")
ggplot(offense) +
aes(
x = `Yards Per Game`,
y = `Points Per Game`,
color = Level
) +
labs(
title= "Points Per Game versus Yards Per Game by Level of Competition"
)+
geom_point(shape = "circle", size = 1.5) +
scale_color_hue(direction = 1) +
theme_minimal()
ggplot(offense) +
aes(
x = `Passing Yards Per Game`,
y = `Points Per Game`,
color = Level
) +
labs(
title= "Points Per Game versus Passing Yards Per Game by Level of Competition"
)+
geom_point(shape = "circle", size = 1.5) +
scale_color_hue(direction = 1) +
theme_minimal()
ggplot(offense) +
aes(
x = `Rushing Yards Per Game`,
y = `Points Per Game`,
color = Level
) +
labs(
title= "Points Per Game versus Rushing Yards Per Game by Level of Competition"
)+
geom_point(shape = "circle", size = 1.5) +
scale_color_hue(direction = 1) +
theme_minimal()
offenseModel <- lm(`Points Per Game` ~ `Yards Per Game`+IndLevel+`Yards Per Game`*IndLevel, data = offense)
b0 <- summary(offenseModel)$coeff[1,1]
b1 <- summary(offenseModel)$coeff[2,1]
b2 <- summary(offenseModel)$coeff[3,1]
b3 <- summary(offenseModel)$coeff[4,1]
gf_point(`Points Per Game` ~ `Yards Per Game`, color=~Level, data = offense) %>%
gf_abline(intercept = b0, slope = b1, color = "red") %>%
gf_abline(intercept = b0+b2, slope = b1+b3, color = "blue")
passingOffenseModel <- lm(`Points Per Game` ~ `Passing Yards Per Game`+IndLevel+`Passing Yards Per Game`*IndLevel, data = offense)
pb0 <- summary(passingOffenseModel)$coeff[1,1]
pb1 <- summary(passingOffenseModel)$coeff[2,1]
pb2 <- summary(passingOffenseModel)$coeff[3,1]
pb3 <- summary(passingOffenseModel)$coeff[4,1]
gf_point(`Points Per Game` ~ `Passing Yards Per Game`, color=~Level, data = offense) %>%
gf_abline(intercept = pb0, slope = pb1, color = "red") %>%
gf_abline(intercept = pb0+pb2, slope = pb1+pb3, color = "blue")
rushingOffenseModel <- lm(`Points Per Game` ~ `Rushing Yards Per Game`+IndLevel+`Rushing Yards Per Game`*IndLevel, data = offense)
rb0 <- summary(rushingOffenseModel)$coeff[1,1]
rb1 <- summary(rushingOffenseModel)$coeff[2,1]
rb2 <- summary(rushingOffenseModel)$coeff[3,1]
rb3 <- summary(rushingOffenseModel)$coeff[4,1]
gf_point(`Points Per Game` ~ `Rushing Yards Per Game`, color=~Level, data = offense) %>%
gf_abline(intercept = rb0, slope = rb1, color = "red") %>%
gf_abline(intercept = rb0+rb2, slope = rb1+rb3, color = "blue")
totalYardsANCOVAModel <- lm(`Points Per Game` ~ `Yards Per Game` + Level, data=offense)
anova(totalYardsANCOVAModel)
passingYardsANCOVAModel <- lm(`Points Per Game` ~ `Passing Yards Per Game` + Level, data=offense)
anova(passingYardsANCOVAModel)
rushingYardsANCOVAModel <- lm(`Points Per Game` ~ `Rushing Yards Per Game` + Level, data=offense)
anova(rushingYardsANCOVAModel)